Kiss those Math Headaches GOODBYE!

Recent insight on the GCF (and GPGCF)

A while back I wrote a post about the GCF, and mentioned that there’s a number  related to it — a number that I call the GPGCF. “GPGCF” stands for the “Greatest Possible Greatest Common Factor.”

In short, the GPGCF is a number that sets an upper limit for the size of the GCF. I’ve seen many students struggle when searching for the GCF, seeking hither and yon for it. I had a sense that students were checking numbers that were too large. That’s what led me to try to figure out what must the the upper limit for the GCF.

If you check out that post (10/25/10), you’ll see that, for any two numbers, I said that the difference between those numbers has to be the GPGCF.

And I was correct, to a degree.

But I recently realized that my little theory needs modifying.

For while the difference between any two numbers can be the upper limit for the GCF, that difference is not the only quantity that can set an upper limit for the GCF. There’s another quantity that plays a role.

That other quantity, I recently realized, is the size of the smaller of the two numbers.

Take the numbers 8 and 24, for example.

The difference between these two numbers is 16, so I would have said that 16 is the upper limit for the GCF. But there’s actually another quantity that limits the size of the GCF, and that quantity is 8. For since the GCF of 8 and 24 must by definition fit into both 8 and 24, it must fit into 8. And common sense tells us that there’s no number larger than 8 that can fit into 8! So the size of this number — the smaller of the two numbers — also sets an upper limit for the size of the GCF.

So my revised theory about the GPGCF is this:  when you need to find the GCF for any two numbers, look at two quantities:  1) the smaller of the two numbers, and 2) the difference between the two numbers. Both of these quantities constrains the size of the GPGCF. So therefore, whichever of these is smaller IS the GPGCF. Once you’ve found the GPGCF, that makes it easier to find the actual GCF.

I know this sounds very abstract, so let’s look at a few examples to see what I’m blabbering on about.

Example 1:  What’s the GPGCF for 6 and 16?

Smaller number is 6; difference is 10.
6 and 10 both limit the size of the GCF, but
6 is less than 10, so 6 is the GPGCF.

Example 2:  What’s the GPGCF for 8 and 12?

Smaller number is 8; difference is 4.
4 is less than 8, so 4 is the GPGCF.

Example 3:  What’s the GPGCF for 30 and 75?

Smaller number is 30; difference is 45.
30 is less than 45, so 30 is the GPGCF.

Example 4:  What’s the GPGCF for 28 and 42?

Smaller number is 28; difference is 14.
14 is less than 28, so 14 is the GPGCF.

Now, let’s go one step further. From here, how do we figure out the GCF? I’ve done a bit more thinking about this, too, and I’ll share those ideas in my next post.

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Place Value Metaphor

During the summer I get to tutor a lot of elementary age students, remediating them on the basics.

Almost invariably I find that these students are confused about PLACE VALUE, and considering how critical this concept is to all of math, I decided to write this post.

Whenever I have the least suspicion that a student might be confused about place value, I check with a simple test.

I have them write down the number 22, then I ask them if they can tell me the difference between the two 2s. Often they cannot.

Tutoring a girl this past week I came up with a way of understanding place value that really resonated with the student. I want to share it because you may be able to use it, or a modification of it, with your students. First it’s important to know that this student’s mom teaches ballet, and the girl dances at her mom’s studio.

I asked the girl if she has ever been to a ballet performance, and of course she said yes.

Then I drew a quick diagram of the stage and first few audience rows. I pointed to two seats, one in the front row, another seat several rows back. I asked her if the two seats would cost the same amount. This girl knew that the close seat costs more money because it is closer to the action on stage.

Then I used that idea to explain place value. I showed this girl that just as seats can be more or less valuable because of where they are, so too digits can be more or less value based on where they are in a number.

She got this idea very quickly, and now she understands place value.

For children with different interests, use whatever makes sense. For example if you’re teaching a boy who loves baseball, make the rows of seats those at a baseball game, and so on.